17.7: Partition Functions of Indistinguishable Molecules

Breakdown of Boltzmann statistics

From quantum mechanics, it can be shown that the above reasoning is only valid if the number of available states is much larger than the number of particles. For that to be the case the following relationship should hold:

\[ \dfrac{N}{V} \left( \dfrac{h^2}{8mk_BT} \right)^{3/2} << 1\]

If this inequality is not fulfilled a different type of statistical distribution needs to be applied. Table 17.1 in the book gives an idea when this happens. The electron gas inside a metal is a clear example of breakdown. Of the physical gases only the lightest two start to deviate a little at low temperatures.